For let A C (in the 10th figure) be the time in which a body is transmitted with uniform motion from A to B; and the parallelogram A B D C being completed, let it be required to find out a length in which that body may be moved in the same time A C from A, with motion so accelerated, that the proportion of the lengths transmitted to that of the times be continually as 3 to 2. Let B D be so divided in E, that B D be to B E as 3 to 2; and between B D and B E let B F be a mean proportional; and let A F be drawn and produced till it meet with C D produced in G; and making A M a mean proportional between A H and A B, let it be as A M to A B, so A B to A I; and so the proportion of A H to A I will be to that of A H to A B as 3 to 2; for of the proportions, of which that of A H to A M is one, that of A H to A B is two, and that of A H to A I is three; and consequently, as 3 to 2 to that of G H to B F, and (F K being drawn parallel to B I and cutting A D in K) so likewise to that of G H or B D to I K. Wherefore the proportion of the length A H to A I is to the proportion of the time B D to I K as 3 to 2; and therefore if in the time A C the body be moved with accelerated motion, as was propounded, till it acquire the impetus H G equal to A C, the length transmitted in the same time will be A H. 16. But if the proportion of the lengths to that of the times had been as 4 to 3, there should then have been taken two mean proportionals between A H and A B, and their proportion should have been continued one term further, so that A H to A B might have three of the same proportions, of which A H to A I has four; and all things else should have been done as is already shown. Now the way how to interpose any number of means between two lines given, is not yet found out. Nevertheless this may stand for a general rule; _if there be a time given, and a length be transmitted in that time with uniform motion; as for example, if the time be_ A C, _and the length_ A B, _the strait line_ A G, _which determines the length_ C G _or_ A H, _transmitted in the same time_ A C _with any accelerated motion, shall so cut_ B D _in_ F, _that_ B F _shall be a mean proportional between_ B D _and_ B E, B E _being so taken in_ B D, _that the proportion of length to length be everywhere to the proportion of time to time, as the whole_ B D _is to its part_ B E. 17. If in a given time two lengths be transmitted, one with uniform motion, the other with motion accelerated in any proportion of the lengths to the times; and again, in part of the same time, parts of the same lengths be transmitted with the same motions, the whole length will exceed the other length in the same proportion in which one part exceeds the other part.